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A multiresolution space-time adaptive scheme for the bidomain model in electrocardiology. (English) Zbl 1206.92004
Summary: The bidomain model of electrical activity of myocardial tissue consists of a possibly degenerate parabolic PDE coupled with an elliptic PDE for the transmembrane and extracellular potentials, respectively. This system of two scalar PDEs is supplemented by a time-dependent ODE modeling the evolution of the gating variable. In the simpler subcase of the monodomain model, the elliptic PDE reduces to an algebraic equation. Since typical solutions of the bidomain and monodomain models exhibit wavefronts with steep gradients, we propose a finite volume scheme enriched by a fully adaptive multiresolution method, whose basic purpose is to concentrate computational effort on zones of strong variation of the solution. Time adaptivity is achieved by two alternative devices, namely locally varying time stepping and a Runge-Kutta-Fehlberg-type adaptive time integration. A series of numerical examples demonstrates that these methods are efficient and sufficiently accurate to simulate the electrical activity in myocardial tissue with affordable effort. In addition, the optimal choice of the threshold for discarding nonsignificant information in the multiresolution representation of the solution is addressed, and the numerical efficiency and accuracy of the method is measured in terms of CPU time speed-up, memory compression, and errors in different norms.

MSC:
92C05 Biophysics
35Q92 PDEs in connection with biology, chemistry and other natural sciences
65N99 Numerical methods for partial differential equations, boundary value problems
92C30 Physiology (general)
65C20 Probabilistic models, generic numerical methods in probability and statistics
92C50 Medical applications (general)
37N25 Dynamical systems in biology
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
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References:
[1] Hodgkin, A quantitative description of membrane current and its application to conduction and excitation in nerve, J Physiol 117 pp 500– (1952) · doi:10.1113/jphysiol.1952.sp004764
[2] Keener (2009)
[3] L.Tung, A bi-domain model for describing ischemic myocardial D-C currents, PhD thesis, MIT, Cambridge, MA, 1978.
[4] Bendahmane, Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue, Netw Heterog Media 1 pp 185– (2006) · Zbl 1179.35162 · doi:10.3934/nhm.2006.1.185
[5] Colli Franzone, Adaptivity in space and time for reaction-diffusion systems in electro-cardiology, SIAM J Sci Comput 28 pp 942– (2006)
[6] Colli Franzone, Degenerate evolution systems modeling the cardiac electric field at micro- and macroscopic level, Evolution equations, semigroups and functional analysis pp 49– (2002) · Zbl 1036.35087 · doi:10.1007/978-3-0348-8221-7_4
[7] Domingues, An adaptive multiresolution scheme with local time-stepping for evolutionary PDEs, J Comput Phys 227 pp 3758– (2008) · Zbl 1139.65060
[8] Cohen, Fully adaptive multiresolution finite volume schemes for conservation laws, Math Comp 72 pp 183– (2001)
[9] Roussel, A conservative fully adaptive multiresolution algorithm for parabolic PDEs, J Comput Phys 188 pp 493– (2003) · Zbl 1022.65093
[10] Bendahmane, Adaptive multiresolution schemes with local time stepping for two-dimensional degenerate reaction-diffusion systems, Appl Numer Math 59 pp 1668– (2009) · Zbl 1400.65042
[11] Bürger, Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension, M2AN Math Model Numer Anal 42 pp 535– (2008)
[12] Bürger, Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux, J Engrg Math 60 pp 365– (2008) · Zbl 1137.65393
[13] Cherry, Efficient simulation of three-dimensional anisotropic cardiac tissue using an adaptive mesh refinement method, Chaos 13 pp 853– (2003) · Zbl 1080.92513
[14] Chen, Global existence and blow-up for the solutions to nonlinear parabolic/elliptic system modelling chemotaxis, IMA J Appl Math 70 pp 221– (2005) · Zbl 1103.35039
[15] Bourgault, Existence and uniqueness of the solution for the bidomain model used in cardiac electro-physiology, Nonlin Anal Real World Appl 10 pp 458– (2009)
[16] Quan, Efficient integration of a realistic two-dimensional cardiac tissue model by domain decomposition, IEEE Trans Biomed Eng 45 pp 372– (1998)
[17] Sundnes, An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso, Math Biosci 194 pp 233– (2005) · Zbl 1063.92018
[18] Skouibine, A numerically efficient model for simulation of defibrillation in an active bidomain sheet of myocardium, Math Biosci 166 pp 85– (2000) · Zbl 0963.92019
[19] Berger, Adaptive mesh refinement for hyperbolic partial differential equations, J Comput Phys 53 pp 482– (1984) · Zbl 0536.65071
[20] Colli Franzone, A parallel solver for reaction-diffusion systems in computational electro-cardiology, Math Models Meth Appl Sci 14 pp 883– (2004) · Zbl 1068.92024
[21] Saleheen, A new three-dimensional finite-difference bidomain formulation for inhomogeneous anisotropic cardiac tissues, IEEE Trans Biomed Eng 45 pp 15– (1998)
[22] Harten, Multiresolution algorithms for the numerical solution of hyperbolic conservation laws, Comm Pure Appl Math 48 pp 1305– (1995) · Zbl 0860.65078 · doi:10.1002/cpa.3160481201
[23] Müller, Adaptive Multiscale Schemes for Conservation Laws (2003) · Zbl 1016.76004 · doi:10.1007/978-3-642-18164-1
[24] Chiavassa, Adaptive mesh refinement-theory and applications pp 137– (2003)
[25] Dahmen, Multiresolution schemes for conservation laws, Numer Math 88 pp 399– (2001) · Zbl 1001.65104
[26] Müller, Fully adaptive multiscale schemes for conservation laws employing locally varying time stepping, J Sci Comput 30 pp 493– (2007) · Zbl 1110.76037
[27] Sundnes, Computing the electrical activity in the heart (2006)
[28] Mitchell, A two-current model for the dynamics of cardiac membrane, Bull Math Biol 65 pp 767– (2001)
[29] Colli Franzone, Simulating patterns of excitation, repolarization and action potential duration with cardiac bidomain and monodomain models, Math Biosci 197 pp 35– (2005) · Zbl 1074.92004
[30] Johnston, The effect of simplifying assumptions in the bidomain model of cardiac tissue: Application to ST segment shifts during partial ischaemia, Math Biosci 198 pp 97– (2005) · Zbl 1076.92028
[31] Bendahmane, Convergence of a finite volume scheme for the bidomain model of cardiac tissue, Appl Numer Math 59 pp 2266– (2009) · Zbl 1165.92005
[32] Y.Coudière,C.Pierre, and R.Turpault, Solving the fully coupled heart and torso problems of electro cardiology with a 3D discrete duality finite volume method. HAL preprint ( 2006), available from http://hal.archives-ouvertes.fr/ccsd-00016825.
[33] M.Bendahmane,R.Bürger, and R.Ruiz-Baier, A finite volume scheme for cardiao propagation in media with isotropic conductivites. Preprint, Universidad de Concepción; submitted. · Zbl 1192.92002
[34] Pennacchio, Efficient algebraic solution of reaction-diffusion systems for the cardiac excitation process, J Comput Appl Math 145 pp 49– (2002) · Zbl 1006.65102
[35] Moore, An adaptive finite element method for parabolic differential systems: some algorithmic considerations in solving in three space dimensions, SIAM J Sci Comput 21 pp 1567– (2000) · Zbl 0969.65090
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